Functionals of Squared Bessel Processes

Abstract

In this chapter, scalar- and multidimensional processes based on the squared Bessel process are discussed and subsequently applied in the context of the benchmark approach. In a first section, results from the literature regarding the squared Bessel process and related processes, namely the Bessel process, the square-root process, the 3/2 process and the CEV process are recalled. In the second part of the chapter, we introduce the Wishart process as the multidimensional generalization of the squared Bessel process. We remark that this process is revisited in Chap. 11, which is entirely devoted to this process. The chapter concludes by recalling from the literature a potential model for the Growth Optimal Portfolio, namely the Minimal Market Model. Formulated under the benchmark approach, the Minimal Market Model employs a squared Bessel process to model the Growth Optimal Portfolio. We demonstrate that an equivalent risk neutral probability measure does not exist for this model, yet the benchmark approach can still accommodate it. Furthermore, we demonstrate that the benchmark approach produces the same level of tractability as the Black-Scholes model. This chapter demonstrates that the benchmark approach can accommodate realistic, yet tractable models for the Growth Optimal Portfolio.